Harmonic Algebraic Curves and Noncrossing Partitions

Motivated by Gauss’s first proof of the Fundamental Theorem of Algebra, we study the topology of harmonic algebraic curves. By the maximum principle, a harmonic curve has no bounded components; its topology is determined by the combinatorial data of a noncrossing matching. Similarly, every complex polynomial gives rise to a related combinatorial object that we call a basketball, consisting of a pair of noncrossing matchings satisfying one additional constraint. We prove that every noncrossing matching arises from some harmonic curve, and deduce from this that every basketball arises from some polynomial.

• Type: Journal
• Year: 2007
• Where: DCG
• Authors: Jeremy L. Martin, David Savitt, Ted Singer
• Keywords: DCG 2007, Gauss’s first Proof, Harmonic Algebraic Curves, Harmonic Curve